Two-generator solution

Introduction

Two-generator is the group of situations which can be reached by only using moves of two connecting sides of the cube, f.i. R and U.
So when scrambling and solving you can only use the moves R, R', R2, U, U' and U2.
The size of this group is 73,483,200 situations. The maximum depth of this group is 25 quarter turns. See also Mark Longridge pages.
With some practice it should be possible to solve the two-generator within 10 seconds on average.

Solution

A good system to solve the two-generator group is the following (assuming the R and U group):
A) solve edge RD
B) solve the two corner/edge pairs FDR+FR and BDR+BR
C) orient upper layer corners (permutation is always correct)
D) permute last layer edges (orientation is always correct)

Below you can find the algorithms that I use to solve the two-generator group, but you can also use my CubeSolver applet to find other algorithms for the cases.

Stage A
This stage is trivial. The RD edge always has the same orientation, so you can move it into the right slice and then do a move of the R side to get it in place.

Stage B
This is the hardest part of the solution. If you are familiar with the F2L system (see also my F2L page), then you just need to practice using only algorithms with R and U moves. In fact in that case solving two-generator may be a good practice to solve pairs at BR (or BL in the L and U group).
Many of the normal F2L cases are not possible because of the fact that the edges cannot be flipped.

Stage C
For this stage there are 7 different cases where the corners are not oriented correctly.

Case id	Case	Algorithm
H1		H5 U' H4
H2		U' H5 U H4
H3		RUR'U RU'R'U RU'R'U RU2R' or RU'RU'R2UR2UR2UR'U'RUR
H4		RUR'URU2R'
H5		RU2R'U'RU'R'
H6		RU2R2U'R2U'R2U2R
H7		RUR'URU'R'URU2R'

Stage D
For this stage there are 4 different cases where the edges are not permuted correctly.

Case id	Case	Algorithm
P4		RUR'U R'U'R'U RU'R'U' R2URU2 or R2UR2UR'U2R2U2R2U2R'U'R2U'R2
P5		R2U2 R2U2 R2U R2U2 R2U2 R2U'
P6		R'U2R UR'UR U RU2R' U'RU'R' U' or R2U'R'U'R2URURU2R
P7		RU2R' U'RU'R' U' R'U2R UR'UR U or R2URUR2U'R'U'R'U2R'

Good luck!